Structures of wing-tip vortices at low Reynolds numbers and their modelling

Abstract

The structures of wing-tip vortices at a low Reynolds number () are studied by direct numerical simulation. After confirming the basic features of the wing-tip vortices, the distributions of axial vorticity and velocity are investigated in detail. The vorticity distribution consists of a dominant symmetric component, quadrupole components responsible for elliptical deformation of the vortex core and the other small components. The symmetric component of vorticity is in reasonable agreement with the Moore-Saffman model (Moore & Saffman 1973 Proc. R. Soc. Lond. 333, 491) for the three types of wing planform considered: the rectangular, elliptic and delta wings. The agreement between numerical results and the Moore-Saffman model is not good for the symmetric component of axial velocity since the boundary layer component is small owing to the incomplete roll-up of the vortex sheet shed from the boundary layer. However, the vortex component of the axial velocity is correctly predicted by the Moore-Saffman model. The elliptical deformation of the vortex core is characterized by the aspect ratio and the direction of the major axis. It is found that the elliptical deformation is not uniform in the streamwise direction, which implies that the wing-tip vortices cannot be always modelled by a two-dimensional vortex pair. The evolution of the elliptical deformation along the streamwise direction is shown to be described by two-dimensional vortex dynamics under the boundary-layer approximation. The equations of motion of an elliptic vortex patch in a straining field predicts correctly the rate of change of the direction, but fails to predict that of the aspect ratio, for which the equation derived for axisymmetrization of vortices gives a good prediction.